In order to estimate the approximation error for using a taylor polynomial for a certain function, I am using the taylor remainder estimation theorem.
where x is the value that is being approximated, a is the center, and M is an upper bound for f (n+1)(x) derivative for all numbers between a and x.
I have the second degree Maclaurin series for ex (1+x+x2/2!) and am trying to approximate e3.
Using what we know, we have that x=3, n=2, and a=0. To find M, we use the f (n+1)(x) derivative and find an upper bound on the interval between a and x. f3(x)=ex, and the highest f3(x) can be on the interval [0,3] is e3, so an upper bound for M can be 20.2 since e3 is around 20.0855.
Plugging this all in we get 20.2(33) / 3! = 20.2(27) / 6 = 90.9.
However, the actual error for using the second degree Maclaurin series to approximate e3 is 10.0855.
Where am I going wrong?
|Rn(x)| ≤ M|x-a|n+1 / (n + 1)!
where x is the value that is being approximated, a is the center, and M is an upper bound for f (n+1)(x) derivative for all numbers between a and x.
I have the second degree Maclaurin series for ex (1+x+x2/2!) and am trying to approximate e3.
Using what we know, we have that x=3, n=2, and a=0. To find M, we use the f (n+1)(x) derivative and find an upper bound on the interval between a and x. f3(x)=ex, and the highest f3(x) can be on the interval [0,3] is e3, so an upper bound for M can be 20.2 since e3 is around 20.0855.
Plugging this all in we get 20.2(33) / 3! = 20.2(27) / 6 = 90.9.
However, the actual error for using the second degree Maclaurin series to approximate e3 is 10.0855.
Where am I going wrong?